Comparison of random and deterministic

2012 International Symposium on Nonlinear Theory and its Applications NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Comparison of random and deterministic characteristics of chaotic signals issued from a one-dimensional piecewise linear map † ‡

Kais Feltekh

‡

†

, Danièle Fournier-Prunaret , Safya Belghith

and Zouhair Ben Jemaa

†

†Syscom Laboratory, ENIT 1002, Tunis, Tunisia ‡LAAS-INSA, University of Toulouse 135 Av. de Rangueil, F-31077 Toulouse, Cedex 4, France Email: [email protected], [email protected], [email protected], [email protected] Abstract This paper presents the characteristics of the power spectral density (PSD) of chaotic signals generated by a one-dimensional piecewise linear map. The majority of previous research on chaos show that chaotic signals are rather broadband with impulsive auto-correlation sequence (ACS). However, recent studies of the skew tent map [4, 5] have shown that the PSD and ACS are modied according to certain values of the bifurcations. We propose to extend this work to other maps and to study relations between bandwidth and Lyapunov exponents.

to obtain chaotic signals with dierent types of bandwidth, using signals obtained from a discrete-time dynamic system. We have chosen a piecewise linear map depending on parameters for which previous studies have shown that chaotic signals can be observed [3]. 2. One-dimensional piecewise linear map with two parameters

We will be more particularly interested in a onedimensional map depending on two parameters, modeled under the following form:

1. Introduction

The chaotic signals are aperiodic signals and have a sensitivity to initial conditions (SIC) [1]. The sensitivity to initial conditions means that the signals obtained with similar initial conditions can become very dierent when the number of iterations tends to innity. Usually in the existing publications the chaotic signals are characterized by an impulsive auto-correlation sequence (ACS) and they have a broadband frequency [6, 9, 12]. Using this property, dierent applications have been considered with chaotic signals. Anyhow, it is possible to construct chaotic signals with dierent kinds of bandwidth, this is one of our purposes in this paper. So, chaotic signals could be used for other applications than those implying broadband frequency. The spectral analysis of signals is used in several domains to extract information and verify the distribution of energy or power in the frequency range [10]. Spectral models are used in many processes of modulation, voice processing, compression and voice recognition [7]. In medicine, the spectral analysis of electrocardiograms and electroencephalograms may provide useful information in diagnostics [10]. Spectral analysis of signals is also crucial in the eld of telecommunications. The specters must be known and clearly dened, due to insucient frequencies available for communication and limited frequency bands in guided media. In this paper, we propose to study the possibility

xn+1 = f (xn , k, φ),

(1)

with n ∈ ℵ; xn ∈ I interval of